Suppose with
. Define
.
Show that is finite for all
and
.
Proof. We consider the equivalent form of as follows (this is nothing but
)
.
We firstly consider the case when . From the identity
if , then
which implies
. Therefore
.
Since provided
and
for
and
, then we have
Thus, is bounded from above for all
. For
, consider
. More precisely,
where . Note that
.
If , then
. Thus,
is bounded from above in
which implies that
is bounded from below in
. Similar, we can prove that
is bounded from above in
.
Finally, from the above estimates, clearly is of class
, and thus, so is
.